Copied to
clipboard

?

G = (C2×D4).303D4  order 128 = 27

56th non-split extension by C2×D4 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: (C2×D4).303D4, C2.16(D4○D8), (C2×Q8).238D4, C2.16(Q8○D8), C4⋊C4.396C23, (C2×C4).296C24, (C2×C8).145C23, (C2×D4).83C23, C23.247(C2×D4), (C2×Q8).71C23, C2.D8.84C22, C4.Q8.13C22, C23.24D4⋊9C2, C4⋊D4.24C22, C22.D8⋊14C2, C22⋊C8.18C22, M4(2)⋊C4⋊28C2, C22⋊Q8.24C22, C23.36D4⋊11C2, C23.20D4⋊15C2, C23.19D4⋊15C2, C23.48D4⋊14C2, (C22×C8).148C22, C22.556(C22×D4), D4⋊C4.159C22, (C22×C4).1012C23, Q8⋊C4.151C22, C4.61(C22.D4), (C2×M4(2)).78C22, C23.33C23⋊10C2, C42⋊C2.125C22, C22.31C24.8C2, C22.23(C22.D4), (C2×C2.D8)⋊27C2, C4.106(C2×C4○D4), (C2×C4).491(C2×D4), (C22×C8)⋊C2⋊10C2, (C2×C4).298(C4○D4), (C2×C4⋊C4).612C22, (C2×C4○D4).141C22, C2.61(C2×C22.D4), SmallGroup(128,1830)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — (C2×D4).303D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C42⋊C2 — C23.33C23 — (C2×D4).303D4
Lower central
C1 — C2 — C2×C4 — (C2×D4).303D4
Upper central
C1 — C22 — C2×C4○D4 — (C2×D4).303D4
Jennings
C1 — C2 — C2 — C2×C4 — (C2×D4).303D4

Subgroups: 380 in 199 conjugacy classes, 92 normal (38 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×11], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×17], D4 [×12], Q8 [×4], C23, C23 [×2], C23, C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×D4 [×3], C2×Q8 [×2], C4○D4 [×8], C22⋊C8 [×4], D4⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×6], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4 [×2], C4⋊D4 [×3], C22⋊Q8 [×2], C22⋊Q8, C22×C8, C2×M4(2), C2×C4○D4 [×2], (C22×C8)⋊C2, C23.24D4, C23.36D4, C2×C2.D8, M4(2)⋊C4, C22.D8 [×2], C23.19D4 [×2], C23.48D4 [×2], C23.20D4 [×2], C23.33C23, C22.31C24, (C2×D4).303D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C22.D4, D4○D8, Q8○D8, (C2×D4).303D4

Generators and relations
 G = < a,b,c,d,e | a2=b4=c2=e2=1, d4=b2, ebe=ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=ab-1, dcd-1=ece=ab2c, ede=ad3 >

Smallest permutation representation
►On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 55)(18 56)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(41 61)(42 62)(43 63)(44 64)(45 57)(46 58)(47 59)(48 60)

(1 43 5 47)(2 60 6 64)(3 45 7 41)(4 62 8 58)(9 17 13 21)(10 52 14 56)(11 19 15 23)(12 54 16 50)(18 37 22 33)(20 39 24 35)(25 63 29 59)(26 48 30 44)(27 57 31 61)(28 42 32 46)(34 53 38 49)(36 55 40 51)

(1 56)(2 23)(3 50)(4 17)(5 52)(6 19)(7 54)(8 21)(9 62)(10 47)(11 64)(12 41)(13 58)(14 43)(15 60)(16 45)(18 25)(20 27)(22 29)(24 31)(26 53)(28 55)(30 49)(32 51)(33 63)(34 48)(35 57)(36 42)(37 59)(38 44)(39 61)(40 46)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(2 28)(3 7)(4 26)(6 32)(8 30)(9 38)(10 14)(11 36)(13 34)(15 40)(17 19)(18 52)(20 50)(21 23)(22 56)(24 54)(27 31)(33 37)(41 57)(42 48)(43 63)(44 46)(45 61)(47 59)(49 55)(51 53)(58 64)(60 62)
 
G:=sub<Sym(64)| (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(41,61)(42,62)(43,63)(44,64)(45,57)(46,58)(47,59)(48,60), (1,43,5,47)(2,60,6,64)(3,45,7,41)(4,62,8,58)(9,17,13,21)(10,52,14,56)(11,19,15,23)(12,54,16,50)(18,37,22,33)(20,39,24,35)(25,63,29,59)(26,48,30,44)(27,57,31,61)(28,42,32,46)(34,53,38,49)(36,55,40,51), (1,56)(2,23)(3,50)(4,17)(5,52)(6,19)(7,54)(8,21)(9,62)(10,47)(11,64)(12,41)(13,58)(14,43)(15,60)(16,45)(18,25)(20,27)(22,29)(24,31)(26,53)(28,55)(30,49)(32,51)(33,63)(34,48)(35,57)(36,42)(37,59)(38,44)(39,61)(40,46), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,28)(3,7)(4,26)(6,32)(8,30)(9,38)(10,14)(11,36)(13,34)(15,40)(17,19)(18,52)(20,50)(21,23)(22,56)(24,54)(27,31)(33,37)(41,57)(42,48)(43,63)(44,46)(45,61)(47,59)(49,55)(51,53)(58,64)(60,62)>;
 
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(41,61)(42,62)(43,63)(44,64)(45,57)(46,58)(47,59)(48,60), (1,43,5,47)(2,60,6,64)(3,45,7,41)(4,62,8,58)(9,17,13,21)(10,52,14,56)(11,19,15,23)(12,54,16,50)(18,37,22,33)(20,39,24,35)(25,63,29,59)(26,48,30,44)(27,57,31,61)(28,42,32,46)(34,53,38,49)(36,55,40,51), (1,56)(2,23)(3,50)(4,17)(5,52)(6,19)(7,54)(8,21)(9,62)(10,47)(11,64)(12,41)(13,58)(14,43)(15,60)(16,45)(18,25)(20,27)(22,29)(24,31)(26,53)(28,55)(30,49)(32,51)(33,63)(34,48)(35,57)(36,42)(37,59)(38,44)(39,61)(40,46), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,28)(3,7)(4,26)(6,32)(8,30)(9,38)(10,14)(11,36)(13,34)(15,40)(17,19)(18,52)(20,50)(21,23)(22,56)(24,54)(27,31)(33,37)(41,57)(42,48)(43,63)(44,46)(45,61)(47,59)(49,55)(51,53)(58,64)(60,62) );
 
G=PermutationGroup([(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,55),(18,56),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(41,61),(42,62),(43,63),(44,64),(45,57),(46,58),(47,59),(48,60)], [(1,43,5,47),(2,60,6,64),(3,45,7,41),(4,62,8,58),(9,17,13,21),(10,52,14,56),(11,19,15,23),(12,54,16,50),(18,37,22,33),(20,39,24,35),(25,63,29,59),(26,48,30,44),(27,57,31,61),(28,42,32,46),(34,53,38,49),(36,55,40,51)], [(1,56),(2,23),(3,50),(4,17),(5,52),(6,19),(7,54),(8,21),(9,62),(10,47),(11,64),(12,41),(13,58),(14,43),(15,60),(16,45),(18,25),(20,27),(22,29),(24,31),(26,53),(28,55),(30,49),(32,51),(33,63),(34,48),(35,57),(36,42),(37,59),(38,44),(39,61),(40,46)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(2,28),(3,7),(4,26),(6,32),(8,30),(9,38),(10,14),(11,36),(13,34),(15,40),(17,19),(18,52),(20,50),(21,23),(22,56),(24,54),(27,31),(33,37),(41,57),(42,48),(43,63),(44,46),(45,61),(47,59),(49,55),(51,53),(58,64),(60,62)])
 

Matrix representation ►G ⊆ GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
1620000
010000
0011030
0001103
0016060
0001606
,
1150000
0160000
0009015
008020
000708
0010090
,
1300000
1340000
00314611
003366
00143143
0014141414
,
100000
1160000
001000
0001600
000010
0000016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,11,0,16,0,0,0,0,11,0,16,0,0,3,0,6,0,0,0,0,3,0,6],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,0,8,0,10,0,0,9,0,7,0,0,0,0,2,0,9,0,0,15,0,8,0],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,3,3,14,14,0,0,14,3,3,14,0,0,6,6,14,14,0,0,11,6,3,14],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
 

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E···4N4O4P4Q8A8B8C8D8E8F
order12222222244444···4444888888
size11112244822224···4888444488

32 irreducible representations

dim11111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○D8Q8○D8
kernel(C2×D4).303D4(C22×C8)⋊C2C23.24D4C23.36D4C2×C2.D8M4(2)⋊C4C22.D8C23.19D4C23.48D4C23.20D4C23.33C23C22.31C24C2×D4C2×Q8C2×C4C2C2
# reps11111122221131822

In GAP, Magma, Sage, TeX 

(C_2\times D_4)._{303}D_4
 
% in TeX
 
G:=Group("(C2xD4).303D4");
 
// GroupNames label
 
G:=SmallGroup(128,1830);
 
// by ID
 
G=gap.SmallGroup(128,1830);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,100,1018,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=e^2=1,d^4=b^2,e*b*e=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a*b^-1,d*c*d^-1=e*c*e=a*b^2*c,e*d*e=a*d^3>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀